Lower bounds of the Laplacian graph eigenvalues

Abstract

In this paper we prove that all positive eigenvalues of the Laplacian of an arbitrary simple graph have some positive lower bounds. For a fixed integer k ⩾ 1 we call a graph without isolated vertices k-minimal if its kth greatest Laplacian eigenvalue reaches this lower bound. We describe all 1-minimal and 2-minimal graphs and we prove that for every k ⩾ 3 the path P k+1 on k + 1 vertices is the unique k-minimal graph.